E27

ELEMENTARY PARTICLES IN PHYSICS 1
Elementary Particles in Physics
S. Gasiorowicz and P. Langacker
Elementary-particle physics deals with the fundamental constituents of mat-ter
and their interactions. In the past several decades an enormous amount of
experimental information has been accumulated, and many patterns and sys-tematic
features have been observed. Highly successful mathematical theories
of the electromagnetic, weak, and strong interactions have been devised and
tested. These theories, which are collectively known as the standard model, are
almost certainly the correct description of Nature, to first approximation, down
to a distance scale 1/1000th the size of the atomic nucleus. There are also spec-ulative
but encouraging developments in the attempt to unify these interactions
into a simple underlying framework, and even to incorporate quantum gravity
in a parameter-free “theory of everything.” In this article we shall attempt to
highlight the ways in which information has been organized, and to sketch the
outlines of the standard model and its possible extensions.
Classification of Particles
The particles that have been identified in high-energy experiments fall into dis-tinct
classes. There are the leptons (see Electron, Leptons, Neutrino, Muonium),
all of which have spin 1
2 . They may be charged or neutral. The charged lep-tons
have electromagnetic as well as weak interactions; the neutral ones only
interact weakly. There are three well-defined lepton pairs, the electron (e−) and
the electron neutrino (νe), the muon (μ−) and the muon neutrino (νμ), and the
(much heavier) charged lepton, the tau (τ), and its tau neutrino (ν ). These
particles all have antiparticles, in accordance with the predictions of relativistic
quantum mechanics (see CPT Theorem). There appear to exist approximate
“lepton-type” conservation laws: the number of e− plus the number of νe mi-nus
the number of the corresponding antiparticles e+ and ¯νe is conserved in
weak reactions, and similarly for the muon and tau-type leptons. These conser-vation
laws would follow automatically in the standard model if the neutrinos
are massless. Recently, however, evidence for tiny nonzero neutrino masses and
subtle violation of these conservations laws has been observed. There is no un-derstanding
of the hierarchy of masses in Table 1 or why the observed neutrinos
are so light.
In addition to the leptons there exist hadrons (see Hadrons, Baryons, Hy-perons,
Mesons, Nucleon), which have strong interactions as well as the elec-tromagnetic
and weak. These particles have a variety of spins, both integral
and half-integral, and their masses range from the value of 135MeV/c2 for the
neutral pion π0 to 11 020MeV/c2 for one of the upsilon (heavy quark) states.
The particles with half-integral spin are called baryons, and there is clear ev-idence
for baryon conservation: The number of baryons minus the number of
antibaryons is constant in any interaction. The best evidence for this is the
stability of the lightest baryon, the proton (if the proton decays, it does so with
a lifetime in excess of 1033 yr). In contrast to charge conservation, there is no

2
Table 1: The leptons. Charges are in units of the positron (e+) charge e =
1.602 × 10−19 coulomb. In addition to the upper limits, two of the neutrinos
have masses larger than 0.05 eV/c2 and 0.005 eV/c2, respectively. The νe, νμ,
and ν are mixtures of the states of definite mass.
Particle Q Mass
e− −1 0.51MeV/c2
μ− −1 105.7MeV/c2
τ− −1 1777MeV/c2
νe 0 0.15 eV/c2
νμ 0 0.15 eV/c2
ν 0 0.15 eV/c2
Table 2: The quarks (spin- 1
2 constituents of hadrons). Each quark carries baryon
3 , while the antiquarks have B = −1
number B = 1
3 .
Particle Q Mass
u (up) 2
3 1.5 − 5MeV/c2
d (down) −1
3 5 − 9MeV/c2
s (strange) −1
3 80 − 155MeV/c2
c (charm) 2
3 1 − 1.4GeV/c2
b (bottom) −1
3 4 − 4.5GeV/c2
t (top) 2
3 175 − 180GeV/c2
deep principle that makes baryon conservation compelling, and it may turn out
that baryon conservation is only approximate. The particles with integer spin
are called mesons, and they have baryon number B = 0. There are hundreds of
different kinds of hadrons, some almost stable and some (known as resonances)
extremely short-lived. The degree of stability depends mainly on the mass of
the hadron. If its mass lies above the threshold for an allowed decay channel,
it will decay rapidly; if it does not, the decay will proceed through a channel
that may have a strongly suppressed rate, e. g., because it can only be driven
by the weak or electromagnetic interactions. The large number of hadrons has
led to the universal acceptance of the notion that the hadrons, in contrast to
the leptons, are composite. In particular, experiments involving lepton–hadron
scattering or e+e− annihilation into hadrons have established that hadrons are
bound states of point-like spin- 1
2 particles of fractional charge, known as quarks.
Six types of quarks have been identified (Table 2). As with the leptons, there
is no understanding of the extreme hierarchy of quark masses. For each type
of quark there is a corresponding antiquark. Baryons are bound states of three
quarks (e. g., proton = uud; neutron = udd), while mesons consist of a quark
and an antiquark. Matter and decay processes under normal terrestrial con-ditions
involve only the e−, νe, u, and d. However, from Tables 2 and 3 we

see that these four types of fundamental particle are replicated in two heavier
families, (μ−, νμ, c, s) and (τ−, ν , t, b). The reason for the existence of these
heavier copies is still unclear.
Classification of Interactions
For reasons that are still unclear, the interactions fall into four types, the elec-tromagnetic,
weak, and strong, and the gravitational interaction. If we take the
proton mass as a standard, the last is 10−36 times the strength of the electromag-netic
interaction, and will mainly be neglected in what follows. (The unification
of gravity with the other interactions is one of the major outstanding goals.)
The first two interactions were most cleanly explored with the leptons, which do
not have strong interactions that mask them. We shall therefore discuss them
first in terms of the leptons.
Electromagnetic Interactions
The electromagnetic interactions of charged leptons (electron, muon, and tau)
are best described in terms of equations of motion, derived from a Lagrangian
function, which are solved in a power series in the fine-structure constant e2/4π~c =
α ≃ 1/137, a small parameter. The Lagrangian density consists of a term that
describes the free-photon field,
L
= −
1
4
Fμ (x)Fμ (x) , (1)
where
Fμ (x) =
∂A(x)
∂xμ −
∂Aμ(x)
∂x (2)
2 [E2(x)−B2(x)] in more common
is the electromagnetic field tensor. L
is just 1
notation. It is written in terms of the vector potential Aμ(x) because the terms
that involve the lepton and its interaction with the electromagnetic field are
simplest when written in terms of Aμ(x):
Ll = i ¯ ψ(x)γ ∂
∂x − ieA(x)ψ(x) − m¯ ψ(x)ψ(x) . (3)
Here ψ(x) is a four-component spinor representing the electron, muon, or tau,
¯ ψ(x) = ψ†(x)γ0, the γ(α = 0, 1, 2, 3) are the Dirac matrices [4 × 4 matrices
that satisfy the conditions (γ1)2 = (γ2)2 = (γ3)2 = −(γ0)2 = −1 and γγ

γ for β6= α]; m has the dimensions of a mass in the natural units in which
~ = c = 1. If e were zero, the Lagrangian would describe a free lepton; with
e6= 0 the interaction has the form
−eA(x)j(x) , (4)
where the current j(x) is given by
j(x) = − ¯ ψ(x)γψ(x) . (5)

4
The equations of motion show that the current is conserved,
∂
∂x
j(x) = 0 , (6)
so that the charge
Q = Z d3r j0(r, t) (7)
is a constant of the motion.
The form of the interaction is obtained by making the replacement
∂
∂x →
∂
∂x − ieA(x) (8)
in the Lagrangian for a free lepton. This minimal coupling follows from a deep
principle, local gauge invariance. The requirement that ψ(x) can have its phase
changed locally without affecting the physics of the lepton, that is, invariance
under
ψ(x) → e−i(x)ψ(x) , (9)
can only be implemented through the introduction of a vector field A(x), cou-pled
as in (8), and transforming according to
A(x) → A(x) −
1
e
∂θ(x)
∂x . (10)
This dictates that the free-photon Lagrangian density contains only the gauge-invariant
combination (2), and that terms of the formM2A2
(x) be absent. Thus
local gauge invariance is a very powerful requirement; it implies the existence
of a massless vector particle (the photon, γ), which mediates a long-range force
[Fig. 1(a)]. It also fixes the form of the coupling and leads to charge conservation,
and implies masslessness of the photon. The resulting theory (see Quantum
Electrodynamics, Compton Effect, Feynman Diagrams, Muonium, Positron) is
in extremely good agreement with experiment, as Table 3 shows. In working
out the consequences of the equations of motion that follow from (3), infinities
appear, and the theory seems not to make sense. The work of S. Tomonaga,
J. Schwinger, R. P. Feynman, and F. J. Dyson in the late 1940s clarified the
nature of the problem and showed a way of eliminating the difficulties. In
creating renormalization theory these authors pointed out that the parameters
e and m that appear in (3) can be identified as the charge and the mass of
the lepton only in lowest order. When the charge and mass are calculated in
higher order, infinite integrals appear. After a rescaling of the lepton fields,
it turns out that these are the only infinite integrals in the theory. Thus by
absorbing them into the definitions of new quantities, the renormalized (i. e.,
physically measured) charge and mass, all infinities are removed, and the rest
of the theoretically calculated quantities are finite. Gauge invariance ensures
that in the renormalized theory the current is still conserved, and the photon
remains massless (the experimental upper limit on the photon mass is 6×10−17
eV/c2).

Fig. 1: (a) Long-range force between electron and proton mediated by a photon.
(b) Four-fermi (zero-range) description of beta decay (n → pe−¯νe). (c) Beta
decay mediated by a W−. (d) A neutral current process mediated by the Z.
Table 3: Extraction of the (inverse) fine structure constant α−1 from various
experiments, adapted from T. Kinoshita, J. Phys. G 29, 9 (2003). The con-sistency
of the various determinations tests QED. The numbers in parentheses
(square brackets) represent the uncertainty in the last digits (the fractional
uncertainty). The last column is the difference from the (most precise) value
α−1(ae) in the first row. A precise measurement of the muon gyromagnetic
ratio aμ is ∼ 2.4σ above the theoretical prediction, but that quantity is more
sensitive to new (TeV-scale) physics.
Experiment Value of −1 Difference from −1(ae)
Deviation from gyromagnetic 137.035 999 58 (52) [3.8 × 10−9] –
ratio, ae = (g − 2)/2 for e−
ac Josephson effect 137.035 988 0 (51) [3.7 × 10−8] (0.116 ± 0.051) × 10−4
h/mn (mn is the neutron mass) 137.036 011 9 (51) [3.7 × 10−8] (−0.123 ± 0.051) × 10−4
from n beam
Hyperfine structure in 137.035 993 2 (83) [6.0 × 10−8] (0.064 ± 0.083) × 10−4
muonium, μ+e−
Cesium D1 line 137.035 992 4 (41) [3.0 × 10−8] (0.072 ± 0.041) × 10−4

6
Subsequent work showed that the possibility of absorbing the divergences of
a theory in a finite number of renormalizations of physical quantities is lim-ited
to a small class of theories, e. g., those involving the coupling of spin- 1
2
to spin-0 particles with a very restrictive form of the coupling. Theories in-volving
vector (spin-1) fields are only renormalizable when the couplings are
minimal and local gauge invariance holds. Thus gauge-invariant couplings like
¯ ψ(x)γγ

(x), which are known not to be needed in quantum electrody-namics,
are eliminated by the requirement of renormalizability. (The apparent
infinities for non-renormalizable theories become finite when the theories are
viewed as a low energy approximation to a more fundamental theory. In that
case, however, the low energy predictions have a very large sensitivity to the
energy scale at which the new physics appears.)
The electrodynamics of hadrons involves a coupling of the form
−eA(x)jhad
(x) . (11)
For one-photon processes, such as photoproduction (e. g., γp → π0p), matrix
elements of the conserved current jhad
(x) are measured to first order in e, while
for two-photon processes, such as hadronic Compton scattering (γp → γp),
matrix elements of products like jhad
(x)jhad

(y) enter. Within the quark theory
one can write an explicit form for the hadronic current:
jhad
(x) =
2
3
¯uγu −
1
3
¯ dγd −
1
3
¯sγs . . . , (12)

where we use particle labels for the spinor operators (which are evaluated at x),
and the coefficients are just the charges in units of e. The total electromagnetic
interaction is therefore −eAj, where
j

=Xi
= j + jhad
Qi ¯ ψiγψi , (13)
and the sum extends over all the leptons and quarks (ψi = e, μ, τ, νe, νμ, ν ,
u, d, c, s, b, t), and where Qi is the charge of ψi.
Weak Interactions
In contrast to the electromagnetic interaction, whose form was already con-tained
in classical electrodynamics, it took many decades of experimental and
theoretical work to arrive at a compact phenomenological Lagrangian density
describing the weak interactions. The form
LW = −
G
√2
J†
(x)J(x) (14)
involves vectorial quantities, as originally proposed by E. Fermi. The current
J(x) is known as a charged current since it changes (lowers) the electric charge
when it acts on a state. That is, it describes a transition such as νe → e− of one

particle into another, or the corresponding creation of an e−¯νe pair. Similarly,
J†
describes a charge-raising transition such as n → p. Equation (14) describes
a zero-range four-fermi interaction [Fig. 1(b)], in contrast to electrodynamics, in
which the force is transmitted by the exchange of a photon. An additional class
of “neutral-current” terms was discovered in 1973 (see Weak Neutral Currents,
Currents in Particle Theory). These will be discussed in the next section. J(x)
consists of leptonic and hadronic parts:
lept(x) + J
J(x) = J
had(x) . (15)
Thus, it describes purely leptonic interactions, such as
μ− → e− + ¯νe + νμ ,
νμ + e− → νe + μ− ,
through terms quadratic in Jlept; semileptonic interactions, most exhaustively
studied in decay processes such as
n → p + e− + ¯νe (beta decay) ,
π+ → μ+ + νμ ,
0 → p + e− + ¯νe ,
and more recently in neutrino-scattering reactions such as
νμ + n → μ− + p (or μ− + hadrons) ,
¯νμ + p → μ+ + n (or μ+ + hadrons) ;
and, through terms quadratic in J
had, purely nonleptonic interactions, such as
0 → p + π− ,
K+ → π+ + π+ + π− ,
in which only hadrons appear. The coupling is weak in that the natural di-mensionless
coupling, with the proton mass as standard, is Gm2
p = 1.01×10−5,
where G is the Fermi constant.
The leptonic current consists of the terms
J
lept(x) = ¯eγ(1 − γ5)νe + ¯μγ(1 − γ5)νμ + ¯τγ(1 − γ5)ν . (16)
Both polar and axial vector terms appear (γ5 = iγ0γ1γ2γ3 is a pseudoscalar
matrix), so that in the quadratic form (14) there will be vector–axial-vector
interference terms, indicating parity nonconservation. The discovery of this
phenomenon, following the suggestion of T. D. Lee and C. N. Yang in 1956 that
reflection invariance in the weak interactions could not be taken for granted but
had to be tested, played an important role in the determination of the phe-nomenological
Lagrangian (14). The experiments suggested by Lee and Yang
all involved looking for a pseudoscalar observable in a weak interaction experi-ment
(see Parity), and the first of many experiments (C. S. Wu, E. Ambler, R.

8
W. Hayward, D. D. Hoppes, and R. F. Hudson) measuring the beta decay of
polarized nuclei (60Co) showed an angular distribution of the form
W(θ) = A + Bpe · hJi , (17)
where pe is the electron momentum and hJi the polarization of the nucleus. The
distribution W(θ) is not invariant under mirror inversion (P) which changes
J → J and pe → −pe, so the experimental form (17) directly showed parity
nonconservation. Experiments showed that both the hadronic and the leptonic
currents had vector and axial-vector parts, and that although invariance under
particle–antiparticle (charge) conjugation C is also violated, the form (14) main-tains
invariance under the joint symmetry CP (see Conservation Laws) when
restricted to the light hadrons (those consisting of u, d, c, and s). There is evi-dence
that CP itself is violated at a much weaker level, of the order of 10−5 of
the weak interactions. As will be discussed later, this is consistent with second-order
weak effects involving the heavy (b, t) quarks, though it is possible that
an otherwise undetected superweak interaction also plays a role. The part of
J
had relevant to beta decay is ∼ ¯uγ(1−γ5)d. The detailed form of the hadronic
current will be discussed after the description of the strong interactions.
Even at the leptonic level the theory described by (14) is not renormalizable.
This manifests itself in the result that the cross section for neutrino absorption
grows with energy:
σ = (const)G2mpE . (18)
While this behavior is in accord with observations up to the highest energies
studied so far, it signals a breakdown of the theory at higher energies, so that
(14) cannot be fundamental. A number of people suggested over the years that
the effective Lagrangian is but a phenomenological description of a theory in
which the weak current J(x) is coupled to a charged intermediate vector boson
W−
(x), in analogy with quantum electrodynamics. The form (14) emerges from
the exchange of a vector meson between the currents (see Feynman Diagrams)
when the W mass is much larger than the momentum transfer in the process
[Fig. 1(c)]. The intermediate vector boson theory leads to a better behaved σ
at high energies. However, massive vector theories are still not renormalizable,
and the cross section for e+e− → W+W− (with longitudinally polarized Ws)
grows with energy. Until 1967 there was no theory of the weak interactions
in which higher-order corrections, though extraordinarily small because of the
weak coupling, could be calculated.
Unified Theories of the Weak and Electromagnetic Inter-actions
In spite of the large differences between the electromagnetic and weak interac-tions
(massless photon versus massive W, strength of coupling, behavior under
P and C), the vectorial form of the interaction hints at a possible common
origin. The renormalization barrier seems insurmountable: A theory involving

vector bosons is only renormalizable if it is a gauge theory; a theory in which a
charged weak current of the form (16) couples to massive charged vector bosons,
LW = −gW[J†(x)W+
(x) + J(x)W−
(x)] , (19)
does not have that property. Interestingly, a gauge theory involving charged
vector mesons, or more generally, vector mesons carrying some internal quantum
numbers, had been invented by C. N. Yang and R. L. Mills in 1954. These
authors sought to answer the question: Is it possible to construct a theory that
is invariant under the transformation
ψ(x) → exp[iT · (x)]ψ(x) , (20)
where ψ(x) is a column vector of fermion fields related by symmetry, the Ti are
matrix representations of a Lie algebra (see Lie Groups, Gauge Theories), and
the θ(x) are a set of angles that depend on space and time, generalizing the
transformation law (9)? It turns out to be possible to construct such a non-
Abelian gauge theory. The coupling of the spin- 1
2 field follows the “minimal”
form (8) in that
∂x ψ → ¯ ψγ ∂
¯ ψγ ∂
∂x + igTiWi
(x) ψ , (21)
where the Wi are vector (gauge) bosons, and the gauge coupling constant g is a
measure of the strength of the interaction. The vector meson form is again
LV = −
1
4
Fμi(x)Fμ
i (x) , (22)
but now the structure of the fields is more complicated than in (2):
Fμi(x) =
∂
∂xμWi
(x) −
∂
∂xWiμ
(x) − gfijkWjμ
(x)Wk
(x) , (23)
because the vector fields Wiμ
themselves carry the “charges” (denoted by the
label i); thus, they interact with each other (unlike electrodynamics), and their
transformation law is more complicated than (10). The numbers fijk that ap-pear
in the additional nonlinear term in (23) are the structure constants of the
group under consideration, defined by the commutation rules
[Ti, Tj] = ifijkTk . (24)
There are as many vector bosons as there are generators of the group. The
Abelian group U(1) with only one generator (the electric charge) is the local
symmetry group of quantum electrodynamics. For the group SU(2) there are
three generators and three vector mesons. Gauge invariance is very restrictive.
Once the symmetry group and representations are specified, the only arbitrari-ness
is in g. The existence of the gauge bosons and the form of their interaction
with other particles and with each other is determined. Yang–Mills (gauge)

10
theories are renormalizable because the form of the interactions in (21) and
(23) leads to cancellations between different contributions to high-energy am-plitudes.
However, gauge invariance does not allow mass terms for the vector
bosons, and it is this feature that was responsible for the general neglect of the
Yang–Mills theory for many years.
S. Weinberg (1967) and independently A. Salam (1968) proposed an ex-tremely
ingenious theory unifying the weak and electromagnetic interactions by
taking advantage of a theoretical development (see Symmetry Breaking, Spon-taneous)
according to which vector mesons in Yang–Mills theories could acquire
a mass without its appearing explicitly in the Lagrangian (the theory without
the symmetry breaking mechanism had been proposed earlier by S. Glashow).
The basic idea is that even though a theory possesses a symmetry, the solutions
need not. A familiar example is a ferromagnet: the equations are rotationally
invariant, but the spins in a physical ferromagnet point in a definite direction.
A loss of symmetry in the solutions manifests itself in the fact that the ground
state, the vacuum, is no longer invariant under the transformations of the sym-metry
group, e. g., because it is a Bose condensate of scalar fields rather than
empty space. According to a theorem first proved by J. Goldstone, this implies
the existence of massless spin-0 particles; states consisting of these Goldstone
bosons are related to the original vacuum state by the (spontaneously broken)
symmetry generators. If, however, there are gauge bosons in the theory, then as
shown by P. Higgs, F. Englert, and R. Brout, and by G. Guralnik, C. Hagen, and
T. Kibble, the massless Goldstone bosons can be eliminated by a gauge trans-formation.
They reemerge as the longitudinal (helicity-zero) components of the
vector mesons, which have acquired an effective mass by their interaction with
the groundstate condensate (the Higgs mechanism). Renormalizability depends
on the symmetries of the Lagrangian, which is not affected by the symmetry-violating
solutions, as was elucidated through the work of B. W. Lee and K.
Symanzik and first applied to the gauge theories by G. ’t Hooft.
The simplest theory must contain a W+ and a W−; since their generators
do not commute there must also be at least one neutral vector boson W0. A
scalar (Higgs) particle associated with the breaking of the symmetry of the
solution is also required. The simplest realistic theory also contains a photon-like
2W
object with its own coupling constant [hence the description as SU(2) × U(1)]. The resulting theory incorporates the Fermi theory of charged-current
weak interactions and quantum electrodynamics. In particular, the vector boson
extension of the Fermi theory in (19) is reproduced with gw = g/2√2, where g
is the SU(2) coupling, and G ≈ √2g2/8M. There are two neutral bosons, the
W0 of SU(2) and B associated with the U(1) group. One combination,
A = cos θWB + sin θWW0 , (25)
is just the photon of electrodynamics, with e = g sin θW. The weak (or Wein-berg)
angle θW which describes the mixing is defined by θW ≡ tan−1(g′/g),
where g′ is the U(1) gauge coupling. In addition, the theory makes the dramatic

prediction of the existence of a second (massive) neutral boson orthogonal to A:
Z = −sin θWB + cos θWW0 , (26)
which couples to the neutral current
JZ
=Xi
T3(i) ¯ ψγ(1 − γ5)ψi − 2 sin2 θWj

, (27)
where j

is the electromagnetic current in (13) and T3(i) [+1
2 for u, ν; −1
2 for
e−, d] is the eigenvalue of the third generator of SU(2). The Z mediates a new
class of weak interactions (see Weak Neutral Currents),
(ν/¯ν) + p, n → (ν/¯ν) + hadrons ,
(ν/¯ν) + nucleon → (ν/¯ν) + nucleon ,
νμ + e− → νμ + e− ,
characterized by a strength comparable to the charged-current interactions [Fig. 1(d)].
Another prediction is that of the existence, in electromagnetic interactions such
as
e− + p → e− + hadrons ,
of tiny parity-nonconservation effects that arise from the exchange of the Z
between the electron and the hadronic system. Neutral current-induced neu-trino
processes were observed in 1973, and since then all of the reactions have
been studied in detail. In addition, parity violation (and other axial current
effects) due to the weak neutral current has been observed in polarized M¨oller
(e−e−) scattering and in asymmetries in the scattering of polarized electrons
from deuterons, in the induced mixing between S and P states in heavy atoms
(atomic parity violation), and in asymmetries in electron–positron annihilation
into μ+μ−, τ+τ−, and heavy quark pairs. All of the observations are in excel-lent
agreement with the predictions of the standard SU(2) × U(1) model and
yield values of sin2 θW consistent with each other. Another prediction is the
existence of massive W± and Z bosons (the photon remains massless because
the condensate is neutral), with masses
M2W
=
A2
sin2 θW
, M2Z
=
M2W
cos2 θW
. (28)
where A ∼ πα/√2G ∼ (37GeV)2. (In practice, a significant, 7%, higher-order
correction must be included.) Using sin2 θW obtained from neutral current
processes, one predicted MW = 80.2±1.1GeV/c2 and MZ = 91.6±0.9GeV/c2’.
In 1983 the W and Z were discovered at the new ¯pp collider at CERN. The
current values of their masses, MW = 80.425 ± 0.038GeV/c2, MZ = 91.1876 ± 0.0021GeV/c2, dramatically confirm the standard model (SM) predictions.
The Z factories LEP and SLC, located respectively at CERN (Switzerland)
and SLAC (USA), allowed tests of the standard model at a precision of ∼ 10−3,

12
Summer 2004
Measurement Fit |Omeas-Ofit|/smeas
0 1 2 3
0 1 2 3
Da(h5a)d(mZ) 0.02761 ± 0.00036 0.02769
m Z [GeV] 91.1875 ± 0.0021 91.1874
G Z [GeV] 2.4952 ± 0.0023 2.4966
shad 0 [nb] 41.540 ± 0.037 41.481
Rl 20.767 ± 0.025 20.739
Afb A0,l 0.01714 ± 0.00095 0.01650
AlA(Pt) (0.1465 ± 0.0032 0.1483
Rb 0.21630 ± 0.00066 0.21562
Rc 0.1723 ± 0.0031 0.1723
Afb A0,b 0.0998 ± 0.0017 0.1040
Afb A0,c 0.0706 ± 0.0035 0.0744
Ab 0.923 ± 0.020 0.935
Ac 0.670 ± 0.026 0.668
AlA(SLD) (0.1513 ± 0.0021 0.1483
sin2qeff lept(Qfb) 0.2324 ± 0.0012 0.2314
m W [GeV] 80.425 ± 0.034 80.394
G W [GeV] 2.133 ± 0.069 2.093
m t [GeV] 178.0 ± 4.3 178.2
Fig. 2: Precision observables, compared with their expectations from the best
SM fit, from The LEP Collaborations, hep-ex/0412015.
much greater than had previously been possible at high energies. The four
LEP experiments accumulated some 2 × 107Z′s at the Z-pole in the reactions
e+e− → Z → ℓ+ℓ− and q¯q. The SLC experiment had a smaller number of
events, ∼ 5 × 105, but had the significant advantage of a highly polarized (∼ 75%) e− beam. The Z pole observables included the Z mass (quoted above),
decay rate, and cross section to produce hadrons; and the branching ratios into
e+e−, μ+μ−, τ+τ− as well as into q¯q, c¯c, and b¯ b. These could be combined to
obtain the stringent constraint N = 2.9841±0.0083 on the number of ordinary
neutrinos with m MZ/2 (i. e., on the number of families with a light ν). The
Z-pole experiments also measured a number of asymmetries, including forward-backward
(FB), polarization, the τ polarization, and mixed FB-polarization,
which were especially useful in determining sin2 θW. The leptonic branching
ratios and asymmetries confirmed the lepton family universality predicted by
the SM. The results of many of these observations, as well as some weak neutral
current and high energy collider data, are shown in Figure 2.
The LEP II program above the Z-pole provided a precise determination of

MW (as did experiments at the Fermilab Tevatron ¯pp collider (USA)), measured
the four-fermion cross sections e+e− → f ¯ f, and tested the (gauge invariance)
predictions of the SM for the gauge boson self-interactions.
The Z-pole, neutral current, and boson mass data together establish that the
standard (Weinberg–Salam) electroweak model is correct to first approximation
down to a distance scale of 10−16 cm (1/1000th the size of the nucleus). In
particular, this confirms the concepts of renormalizable field theory and gauge
invariance, as well as the SM group and representations. The results yield
the precise world average sin2 θW = 0.23149 ± 0.00015. (It is hoped that the
value of this one arbitrary parameter may emerge from a future unification of
the strong and electromagnetic interactions.) The data were precise enough to
allow a successful prediction of the top quark mass (which affected higher order
corrections) before the t was observed directly, and to strongly constrain the
possibilities for new physics that could supersede the SM at shorter distance
scales. The major outstanding ingredient is the Higgs boson, which is hard to
produce and detect. The precision experiments place an upper limit of around
250GeV/c2 on the Higgs mass (which is not predicted by the SM), while direct
searches at LEP II imply a lower limit of 114.4GeV/c2. Some physicists suspect
that the elementary Higgs field may be replaced by a dynamical or bound-state
symmetry-breaking mechanism, but the possibilities are strongly constrained
by the precision data. Unified theories, such as superstring theories, generally
imply an elementary Higgs. It is hoped that the situation will be clarified by
the next generation of high energy colliders.
The Strong Interactions
The strength of the coupling that manifests itself in nuclear forces and in the
interaction of pions with nucleons is such that perturbation theory, so useful
in the electromagnetic interaction, cannot be applied to any field theory of the
strong interactions in which the mesons and baryons are the fundamental fields.
The large number of hadronic states strongly suggests a composite structure
that cannot be viewed as a perturbation about noninteracting systems. In fact,
it is now generally believed that the strong interactions are described by a gauge
theory, quantum chromodynamics (QCD), in which the basic entities are quarks
rather than hadrons. Nevertheless, prior and parallel to the development of the
quark theory a wealth of experimental information concerning the hadrons and
their interactions was accumulated. In spite of the absence of guidance from
field theory, and in spite of the fact that each jump in available accelerator
energy brought a shift in the focus of attention, certain simple patterns were
identified.
Internal Symmetries
The first hint of a new symmetry can be seen in the remarkable resemblance
between neutron and proton. They differ in electromagnetic properties, and,
other than that, by effects that are very small; for example, they differ in mass

14
by 1 part in 700. W. Heisenberg conjectured that the neutron and proton are
two states of a single entity, the nucleon (see Nucleon), just as an electron
with spin up and an electron with spin down are two states of a single entity,
even though in an external magnetic field they have slightly different energies.
Pursuing this analogy, Heisenberg and E. U. Condon proposed that the strong
interactions are invariant under transformations in an internal space, in which
the nucleon is a spinor (see Isospin). Thus, the nucleon is an isospin doublet,
with Iz(p) = 1
2 and Iz(n) = −1
2 , and isospin (in analogy with angular momen-tum)
is conserved. In the language of group theory, the assertion is that the
strong interactions are invariant under the transformations of the group SU(2),
and that particles transform as irreducible representations. The electromagnetic
and weak interactions violate this invariance. The expression for the charge of
the nucleons and antinucleons,
Q = Iz + B/2 , (29)
shows that the charge picks out a preferred direction in the internal space. (It
is now believed that the strong interactions themselves have a small piece which
breaks isospin symmetry, in addition to electroweak interactions.)
With the discovery of the three pions (π+, π0 , π−) with mass remarkably
close to that predicted by H. Yukawa (1935) in his seminal work explaining
nuclear forces in terms of an exchange of massive quanta of a mesonic field, the
notion of isospin acquired a new significance. It was natural, in view of the small
π±−π0 mass difference, to assign the pion to the I = 1 representation of SU(2).
The invariance of the pion–nucleon interaction under isospin transformations led
to a number of predictions, all of which were confirmed. In particular, states
initiated in pion-nucleon collisions could only have isospin 1
2 or 3
2 . Early work
on pion–nucleon scattering led to the discovery of a resonance with rest mass
1236MeV/c2, width 115MeV/c2, and angular momentum and parity JP = 3
2
+
.
This resonance occurred in π+p scattering, so that it had to have I = 3
2 , and
its effects seen in π−p → π−p and π−p → π0n should be the same as those in
π+p → π+p. This prediction was borne out by experiment.
Formally, SU(2) invariance is described by defining generators Ii; (i = 1, 2, 3)
obeying the Lie algebra
[Ii, Ij ] = ieijkIk , (30)
where eijk is totally antisymmetric in the indices and e123 = 1. The statement
that a pion is an I = 1 state then means that the pion field a transforms
according to
[Ii,a] = −(Ii)abb , a = 1, 2, 3 , (31)
where the Ii are 3×3 matrices satisfying (30). In relativistic quantum mechanics
conservation laws must be local, so the conservation law
dIi
dt
= 0 (32)
really follows from the local conservation law
∂
∂xμ Iμ
i (x) = 0 (33)

for the isospin-generating currents, for which
Ii = Z d3r I0
i (r, t) . (34)
Isospin [and SU(3)] are global symmetries: The symmetry transformations are
the same at all space-time points, as opposed to the local (gauge) transforma-tions
in (20). Hence, they are not associated with gauge bosons or a force.
In the early 1950s a number of new particles were discovered. The great
confusion generated by the widely differing rates of production and decay was
cleared up by M. Gell-Mann and K. Nishijima, who extended the notion of
isospin conservation to the strong interactions of the new particles, classified
them (and along the way noted “missing” particles that had to exist, and were
subsequently found), and discovered that the observed patterns of reactions
could be explained by assigning a new quantum number S (strangeness) to each
isospin multiplet.
The selection rules were
S = 0 (35)
for the strong and electromagnetic interactions, and
S = 0, ±1 (36)
for the weak interactions. Relation (29) now takes the form
Q = Iz + (B + S)/2 . (37)
[Equation (37) holds for all hadrons except for those involving heavy (c, b, and
t) quarks, discovered in the 1970s and later.]
The success of the strangeness scheme immediately started a search for a
higher symmetry that would include isospin and strangeness (or hypercharge,
Y = B + S), and that would, in some limit, include the nucleons and the
newly discovered strange baryons in a supermultiplet. The search ended when
M. Gell-Mann and Y. Ne’eman discovered that the Lie group SU(3) was the
appropriate (global) symmetry. The group is generated by eight operators Fi
(i = 1, 2, 3, . . . , 8), of which the first three may be identified with the isospin
generators Ii, and (by convention) F8 is related to hypercharge. The other four
change isospin and strangeness. The nucleons and six other baryons discovered
in the 1950s fit into an eight-dimensional (octet) representation containing dou-blets
with I = 1
2 and Y = ±1, and I = 1, 0 states with Y = 0. Similarly, the
2 , Y = 1, and ( ¯K
I = 1 pions, the (K+,K0) with I = 1
0,K−) with I = 1
2 ,
Y = −1, could be fitted into an octet that was soon completed with the discov-ery
of an I = Y = 0 pseudoscalar meson, the η (see Table 4). SU(3) is only an
approximate symmetry of the strong interactions. Mass splittings within SU(3)
multiplets and other breaking effects are typically 20-30%.
Most interesting is that the search for partners of the resonance (1236) with
I = 3
2 led to a dramatic confirmation of SU(3). The simplest representation
containing an (I = 3
2 , Y = 1) state is the 10-dimensional representation, which

16
Table 4: Table of low-lying mesons and baryons, grouped according to SU(3)
multiplets. There may be considerable mixing between the SU(3) singlets η′,
ϕ, and f′ and the corresponding octet states η, ω, f.
Mass Quark
Particle B Q Y I JP (GeV/c2) content
π 0 1, 0, −1 0 1 0− 0.14 u ¯ d, u¯u − d ¯ d, d¯u
K 0 1, 0 1 1
2 0− 0.49 u¯s, d¯s
¯K
2 0− 0.49 s ¯ d, s¯u
0 0, −1 −1 1
η 0 0 0 0 0− 0.55 u¯u + d ¯ d − 2s¯s
η′ 0 0 0 0 0− 0.96 u¯u + d ¯ d + s¯s
ρ 0 1, 0, −1 0 1 1− 0.77 u ¯ d, u¯u − d ¯ d, d¯u
K∗ 0 1, 0 1 1
2 1− 0.89 u¯s, d¯s
¯K
2 1− 0.89 s ¯ d, s¯u
∗ 0 0, −1 −1 1
ω 0 0 0 0 1− 0.78 u¯u + d ¯ d
φ 0 0 0 0 1− 1.02 s¯s
A2 0 1, 0, −1 0 1 2+ 1.32 u ¯ d, u¯u − d ¯ d d¯u
K∗(1430) 0 1, 0 1 1
2 2+ 1.43 u¯s, d¯s
¯K
2 2+ 1.43 s ¯ d, s¯u
∗(1430) 0 0, −1 −1 1
f 0 0 0 0 2+ 1.28 u¯u + d ¯ d
f′ 0 0 0 0 2+ 1.53 s¯s
N 1 1, 0 1 1
2
1
2
+
0.94 uud, udd
1 0 0 0 1
2
+
1.12 uds − dus
1 1, 0, −1 0 1 1
2
+
1.19 uus, uds + dus, dds
1 0, −1 −1 1
2
1
2
+
1.32 uss, dss
1 2, 1, 0, −1 1 3
2
3
2
+
1.23 uuu, uud, udd, ddd
(1385) 1 1, 0, −1 0 1 3
2
+
1.39 uus, uds, dds
∗(1530) 1 0, −1 −1 1
2
3
2
+
1.53 uss, dss

− 1 −1 −2 0 3
2
+
1.67 sss

also contains (I = 1, Y = 0) and (I = 1
2 , Y = −1) states and an isosinglet Y =
−2 particle. The symmetry-breaking pattern that explained the mass splittings
among the isospin multiplets in the octet predicted equal mass splittings. Thus,
when the I = 1 (1385) was discovered, predictions could be made about the
I = 1
2 ∗, found at mass 1530MeV/c2, and the
−, predicted at 1675MeV/c2.
The latter mass is too low to permit a strangeness-conserving decay to 0K−,
so the
− had to be long-lived, only decaying by a chain of S = 1 weak
interactions with a very clear signature. The dramatic discovery in 1964 of the

− with all the right properties convinced all doubters. [see SU(3) and Higher
Symmetries, Hyperons, Hypernuclear Physics and Hypernuclear Interactions].
S-Matrix Theory
The construction of higher-energy accelerators, the invention of the bubble
chamber by D. Glaser, and the combination of large hydrogen bubble chambers,
rapid scanning facilities, and high-speed computers into a massive data produc-tion
and analysis technology, pioneered by L. Alvarez and collaborators, led to
the discovery of many new resonances during the 1950s and 1960s. The basic
procedure was to measure charged tracks in bubble-chamber pictures, taken in
strong magnetic fields, and to calculate the invariant masses (PEi)2−(Ppic)2
for various particle combinations. Resonances manifest themselves as peaks in
mass distributions, and the events in the resonance region may be further ana-lyzed
to find out the spin and parity of the resonance. Baryonic resonances were
also discovered in phase-shift analyses of angular distributions in pion–nucleon
and K–nucleon scattering reactions. The patterns of masses and quantum num-bers
of the resonances showed that all the mesonic resonances came in SU(3)
octets and singlets, and the baryonic ones in SU(3) decuplets, octets, and sin-glets.
There was good evidence that there was no fundamental distinction between
the stable particles and the highly unstable resonances: The and the
−,
discussed above, are good examples, and theoretically it was found that both
stable (bound) states and resonant ones appeared in scattering amplitudes as
pole singularities, differing only in their location. Furthermore, the role assigned
by Yukawa to the pion as the nuclear “glue” – it was the particle whose exchange
was largely responsible for the nuclear forces – had to be shared with other par-ticles:
Various vector and scalar mesons were seen to contribute to the nuclear
forces, and G. F. Chew and F. E. Low explained much of low-energy pion physics
in terms of nucleon exchange. Chew, in collaboration with S. Mandelstam and
S. Frautschi, proposed to do away with the notion of any particles being “fun-damental.”
They hypothesized that the collection of all scattering amplitudes,
the scattering matrix, be determined by a set of self-consistency conditions, the
bootstrap conditions (see S-Matrix Theory), according to which, crudely stated,
the exchange of all possible particles should yield a “potential” whose bound
states and resonances should be identical with the particles inserted into the
exchange term.
Much effort was devoted to bootstrap and S-matrix theory during the 1960s

18
and early 1970s. The programhad its greatest success in developing phenomeno-logical
models for strong interaction scattering amplitudes at high energies and
low-momentum transfers, such as elastic scattering and total cross sections. In
particular, Mandelstam applied an idea due to T. Regge to relativistic quantum
mechanics, which related a number (perhaps infinite) of particles and resonances
with the same SU(3) and other internal quantum numbers, but different masses
and spins, into a family or Regge trajectory. The exchange of this trajectory of
particles led to much better behaved high-energy amplitudes than the exchange
of one or a small number, in agreement with experiment (see Regge Poles). Re-lated
models had some success in describing inclusive processes (in which one
or a few final particles are observed, with the others summed over) and other
highly inelastic processes (see Inclusive Reactions).
The more ambitious goal of understanding the strong interactions as a boot-strap
(self-consistency) principle met with less success, although a number of
models and approximation schemes enjoyed some measure in limited domains.
The most successful was the dual resonance model pioneered by G. Veneziano.
The dual model was an explicit closed-form expression for strong-interaction
scattering amplitudes which properly incorporated poles for the Regge trajecto-ries
of bound states and resonances that could be formed in the reaction, Regge
asymptotic behavior, and duality (the property that an amplitude could be de-scribed
either as a sum of resonances in the direct channel or as a sum of Regge
exchanges). However, the original simple form did not incorporate unitarity,
i. e., the amplitudes did not have branch cuts corresponding to multiparticle in-termediate
states, and the resonances in the model had zero width (their poles
occurred on the real axis in the complex energy plane instead of being displaced
by an imaginary term corresponding to the resonance width). Perhaps the most
important consequence of dual models was that they were later formulated as
string theories, in which an infinite trajectory of “elementary particles” could
be viewed as different modes of vibration of a one-dimensional string-like ob-ject
(see String Theory). String theories never quite worked out as a model of
the strong interactions, but the same mathematical structure reemerged later
in “theories of everything.”
Many of the S-matrix results are still valid as phenomenologicalmodels. How-ever,
the bootstrap idea has been superseded by the success of the quark theory
and the development of QCD as the probable field theory of the strong interac-tions.
Quarks as Fundamental Particles
The discovery of SU(3) as the underlying internal symmetry of the hadrons and
the classification of the many resonances led to the recognition of two puzzles:
Why did mesons come only in octet and singlet states? Why were there no
particles that corresponded to the simplest representations of SU(3), the triplet
3 and its antiparticle 3∗? M. Gell-Mann and G. Zweig in 1964 independently
proposed that such representations do have particles associated with them (Gell-
Mann named them quarks), and that all observed hadrons are made of (q¯q)

Table 5: The u, d, and s quarks.
B Q Y I Iz
u 1
3
2
3
1
3
1
2
1
2
3 −1
d 1
3
1
3
1
2 −1
2
s 1
3 −1
3 −2
3 0 0
(quark+antiquark) if they have baryon number B = 0 and of (qqq) (three
quarks) if they have baryon number B = 1. They proposed that there exist
three different kinds of quarks, labeled u, d, and s. These were assumed to have
spin 1
2 and the internal quantum numbers listed in Table 5. The quark contents
of the low-lying hadrons are given in Table 4. The vector meson octet (ρ, K∗, ω)
differs from the pseudoscalars (π, K, η) in that the total quark spin is 1 in the
former case and zero in the latter. The (A2, K∗(1490), f) octet are interpreted
as an orbital excitation (3P2). All of the known particles and resonances can
be interpreted in terms of quark states, including radial and orbital excitations
and spin.
The first question was answered automatically, since products of the simplest
representations decompose according to the rules
3 × 3 = 1 + 8 ,
3 × 3 × 3 = 1 + 8 + 8 + 10 .
(38)
A problem immediately arose in that the decuplet to which (1236) belongs,
being the lowest-energy decuplet, should have its three quarks in relative S
states. Thus the ++, whose composition is uuu, could not exist, since the spin-statistics
connection requires that the wave function be totally antisymmetric,
which it manifestly is not when the ++ is in a Jz = 3
2 state, with all spins
up, for example. The solution to this problem, proposed by O. W. Greenberg,
M. Han, and Y. Nambu and further developed by W. A. Bardeen, H. Fritzsch,
and M. Gell-Mann, was the suggestion that in addition to having an SU(3)
label such as (u, d, s) – named flavor by Gell-Mann – and a spin label (up,
down), quarks should have an additional three-valued label, named color. Thus
according to this proposal there are really nine light quarks:
uR uB uY
dR dB dY
sR sB sY
Hence, the low-lying (qqq) state could be symmetric in the flavor and spin labels,
provided it were totally antisymmetric in the color (red, blue, yellow) labels.
More colors could be imagined but at least three are needed. Transformations
among the color labels lead to another symmetry, SU(3)color. The totally anti-symmetric
state is a color singlet. The mesons can also be constructed as color

20
singlets, for example,
π+ =
1
√3
(uR ¯ dR + uB ¯ dB + uY ¯ dY) .
The existing hadronic spectrum shows no evidence for states that could be color
octets, for example, so the present attitude is that either color nonsinglet states
are very massive compared with the low-lying hadrons or that it is an intrinsic
part of hadron dynamics that only color singlet states are observable.
The first evidence that there are three (and not more) colors came from the
study of π0 → 2γ decay. Using general properties of currents, S. Adler and W.
A. Bardeen were able to prove that the π0 decay rate was uniquely determined
by the process in which the π0 first decays into a u¯u or a d ¯ d pair, which then
annihilates with the emission of two photons. The matrix element depends
on the charges of the quarks, and a calculation of the width yields 0.81 eV.
With n colors, this is multiplied by n2, and the observed width of 7.8 ± 0.6 eV
supports the choice of n = 3. Subsequent evidence for three colors was provided
by the total cross section for e+e− annihilation into hadrons (see below), and
by the elevation of the SU(3)color symmetry to a gauge theory of the strong
interactions.
The quark model has been extremely successful in the classification of ob-served
resonances, and even predictions of decay widths work very well, with
much data being correlated in terms of a few parameters. The ingredients that
go into the calculation are (a) that quarks are light, with the (u, d) doublet al-most
degenerate, with mass in the 300-MeV/c2 range (one-third of a nucleon),
(b) that the s quark is about 150MeV/c2 more massive – this explains the pat-tern
of SU(3) symmetry breaking – and (c) that the low-lying hadrons have
the simple q¯q or qqq content, without additional q¯q pairs. However, nobody has
ever observed an isolated quark (free quarks should be easy to identify because
of their fractional charge). It is now generally believed that quarks are con-fined,
i. e., that it is impossible, even in principle, for them to exist as isolated
states. However, in the 1960s this led most physicists to doubt the existence of
quarks as real particles. That view was shattered by the deep inelastic electron
scattering experiments in the late 1960s.
Deep Inelastic Reactions and Asymptotic Freedom
In 1968 the first results of the inelastic electron-scattering experiments (Fig. 3),
e + p → e′ + hadrons
measured at the Stanford Linear Accelerator Center (SLAC), were announced.
The experiments were done in a kinematic region that was new. Both the
momentum transfer squared (that is, the negative mass squared of the virtual
photon exchanged) and the “mass” of the hadronic state produced were large.
The cross section could be written as
d2σ
dE′d

d
point W2(x,Q2) + 2W1(x,Q2) tan2 θ
= dσ
2 , (39)

Fig. 3: Kinematics of deep inelastic lepton scattering.
where (dσ/d
)point is essentially the cross section for a collision with a free point
particle, and the hadronic part of the process was expressed in terms of certain
structure functions W1 and W2. In (39), E′ is the energy of the final electron,
θ the scattering angle, Q2 = −(pe − pe′ )2, and the quantity x is Q2/2mpν,
where ν = p · q/mp is the electron energy loss. No one knew what to expect
for the behavior of W1 and W2. On the one hand, cross sections for production
of definite resonances (exclusive reactions rather than inclusive ones) fell as
powers of 1/Q2; on the other hand, J. D. Bjorken had predicted, on simple
grounds of the irrelevance of masses when all the variables were large, that
the dimensionless functions F1(x,Q2) ≡ mpW1 and F2(x,Q2) ≡ νW2 should
depend on x alone.
The results spectacularly confirmed Bjorken’s conjecture of scaling. R. P.
Feynman interpreted the detailed shapes of the distributions with his parton
model, in which the proton, in a frame in which it is moving rapidly, looks like
a swarm of independently moving point “parts” without any structure. The
shape of F2 can be interpreted as pq2
pxfp(x), where fp(x) is the probability
distribution for a parton to carry a fraction x of the proton’s momentum and
qp is the parton’s charge, while the relation between F1 and F2 depends on the
parton spin. The observed relation F2 ≃ 2xF1 establishes that the partons have
spin 1
2 . Comparing the structure functions obtained from e and μ scattering
(from proton and nuclear targets) with those obtained by weak reactions such
as
νμ(¯νμ)p → μ−(μ+) + hadrons ,
one can constrain the parton quantum numbers. They are consistent with the
assumption that the partons are quarks, and that the proton consists of the three
valence quarks assigned to it by the naive quark model, supplemented with a
sea of quark–antiquark pairs. The relative amount of q¯q sea and its composition

22
Fig. 4: Distributions f(x) times the fraction x of the proton’s momentum carried
by valence quarks uv, dv; gluons g; and sea quarks ¯ d, ¯u, s = ¯s, and c = ¯c, from
S. Eidelman et al. [Particle Data Group], Phys. Lett. B592, 1 (2004).
(e. g., amount of s¯s relative to u¯u) are also determined. The mechanism for deep
inelastic scattering is the ejection of a single quark by the virtual photon, or by
the weak current in the neutrino reactions. The model assumes that the quarks
that make up the proton do not interact, and that seems somewhat mysterious.
Furthermore, the model of the mechanism suggests that one quark is strongly
deflected from the original path. If that is so, where is it?
The problem of how the quarks appear to be noninteracting is answered by
quantum field theory. There we find that the coupling strength is really mo-mentum
dependent. For example, in quantum electrodynamics, because of the
polarizability of the vacuum, the net charge of an electron seen from afar (low
momentum transfer) is smaller than the charge as seen close in (large momen-tum
transfer) where it is not shielded by the positrons produced virtually in the
vacuum. Quantum electrodynamics is not the right kind of theory for quarks,
since the coupling (charge) increases with momentum transfer. It was pointed
out by D. Gross and F. Wilczek, by D. Politzer, and by G. ’t Hooft that a theory
of quarks coupled via Yang–Mills vector mesons will have the property desired
for the quarks probed with high-momentum-transfer currents. The requirement
of such a high-momentum-transfer decoupling, named asymptotic freedom, thus
suggests that the “glue” that binds the quarks together is generated by a non-
Abelian gauge theory, which has the attraction of being renormalizable, uni-versal
(only one coupling constant), and unique, once the number of “colors,”
that is, the group structure, is determined. The high-energy lepton scattering
experiments provide evidence for the existence of some kind of flavor-neutral
glue, in that the data are well fitted in terms of quarks, except that only about
50% of the momentum of the initial proton is attributable to quarks. It is now

Fig. 5: Fundamental QCD interactions. gs is the SU(3)color gauge coupling,
which becomes small at large momentum transfers, and G is a gluon.
understood that the proton momentum is shared by the valence and sea quarks
and electrically neutral gluons. Fig. 4 shows the current experimental situation.
Quantum Chromodynamics
Quantum chromodynamics (QCD), the modern theory of the strong interac-tions,
is a non-Abelian gauge theory based on the SU(3)color group of trans-formations
which relate quarks of different colors. (The transformations are
carried out simultaneously for each flavor, which does not change. The number
of flavors is arbitrary.) The gauge bosons associated with the eight group gener-ators,
known as gluons, can be emitted or absorbed by quarks in transitions in
which the color (but not flavor) can change. Since the gluons themselves carry
color they can interact with each other as well (Fig. 5). As long as there are
no more than 16 flavors, QCD is weakly coupled at large momentum transfers
(asymptotic freedom) and strongly coupled at small momentum transfers, in
agreement with observations.
For the theory to be renormalizable the gluons must either acquire mass
through a Higgs mechanism (spontaneous symmetry breaking) or remain mass-less.
The first type of theory destroys asymptotic freedom (its raison d’ˆetre),
and the second has the difficulty that no massless vector mesons, aside from
the photon, have ever been observed. It has been proposed that the theory
has a structure such that only color singlets are observable, so that the vector
mesons, like the quarks, are somehow confined. It has been speculated that
the non-Abelian field lines, in contrast to electric and magnetic field lines, do
not fan out all over space, but remain confined to a narrow cylindrical region,
which leads to an interaction energy that is proportional to the separation of
the sources of the field lines, and thus confinement. The linearly extended struc-ture
so envisaged is reminiscent of the string models suggested by duality, and
thus may yield the spectrum characteristics of linear Regge trajectories and the
associated high-energy behavior. At large separations the potential presumably
breaks down, with energy converted into (q¯q) pairs, that is, hadrons. This would
explain why quarks are never seen in deep inelastic scattering or other processes.
This picture of quark and gluon confinement has not been rigorously established
in QCD, but is strongly supported by calculations in the most promising approx-imation
scheme for strongly coupled theories: viz., lattice calculations, in which
the space-time continuum is replaced by a discrete four-dimensional lattice (see

24
Fig. 6: One-pion exchange in QCD.
Lattice Gauge Theory).
QCD is very successful qualitatively, but is hard to test quantitatively. This
is partly because the coupling is large for most hadronic processes. Also, QCD
brings a subtle change in perspective. The “strong interactions” are those medi-ated
by the color gluons between quarks, and they give rise to the color singlet
hadronic bound states. The interaction between these states need not be simple,
any more than the interactions between molecules (the van der Waals forces)
manifest the simplicity of the underlying Coulomb force in electromagnetism. It
is hoped, but not conclusively proved, that successful phenomenological models
such as Regge theory or the one-boson-exchange potential emerge as complicated
higher-order effects (Fig. 6). Similarly, it has not been possible to fully calculate
the hadron spectrum (because of strong coupling, relativistic, and many-body
effects), but lattice attempts are promising. Glueballs (bound states of gluons)
and other nonstandard color singlet states are expected. Candidate mesons ex-ist
but have not been unambiguously interpreted. Similar statements apply to
pentaquark states, such as uudd¯s. QCD fairly naturally explains the observed
hadronic symmetries. Parity and CP invariance (except for possible subtle
nonperturbative effects) and the conservation of strangeness and baryon num-ber
are automatic, while approximate symmetries such as isospin, SU(3)flavor,
and chiral symmetry (see below) can be broken only by quark mass terms.
More quantitative tests of QCD are possible in high-momentum-transfer pro-cesses,
in which one glimpses the underlying quarks and gluons. To zeroth order
in the strong coupling gs, QCD reproduces the quark–parton model. Higher-order
corrections lead to calculable logarithmic variations of F1 and F2 with
Q2, in agreement with the data. These experiments have been pushed to much
higher Q2 at the e+p collider HERA at DESY in Hamburg, and the results are
in excellent agreement with QCD.
Another consequence of the quark–parton picture is the prediction that at
high energies the cross section for e+ + e− →hadrons should proceed through
the creation (via a virtual photon) of a q¯q pair, which subsequently converts into
hadrons through the breakdown mechanism (Fig. 7). Thus the cross section is
expected to be point-like, with the modification that the quark charges appear

Fig. 7: Schematic picture of hadron production in e+ + e− annihilation.
at the production end, so that
R ≡
σ(e+ + e− → hadrons)
σ(e+ + e− → μ+ + μ−) ⇒ X all quarks
Q2i
, (40)
where the Qi are the quark charges. At relatively low energies the (u, d, s)
contribution is 2/3 per color, that is, 2. Above the energy (3–4GeV) needed to
produce a charm quark pair (c¯c) one expects R = 10/3, while above the bottom
(b¯ b) threshold (∼ 10GeV), R ∼ 11/3. Calculable higher-order corrections in
QCD increase these predictions slightly. The new contribution of a virtual
Z boson becomes apparent above ∼ 30GeV, with the Z peak dominating at
∼ 90GeV. These predictions are in excellent agreement with the data (Fig. 8),
strongly supporting QCD and the existence of color.
In large-Q2 processes at sufficiently high energies it is expected (and ob-served)
that the produced hadrons tend to cluster in reasonably well-collimated
jets of particles following approximately the direction of the final quarks. For
example, the angular distribution of the jets observed in e+e− annihilation at
SLAC and DESY confirms that the quarks have spin 1
2 , as well as the existence
of the intermediate quark–antiquark state. Similarly, experiments at DESY
have shown the existence of three-jet events whose characteristics are consistent
with the hadronization of a q¯q pair as well as a gluon (gluon bremsstrahlung,
analogous with electron bremsstrahlung). These results give fairly convincing
evidence for the existence of gluons, and in particular establish their spin as 1.
Finally, jets produced in hadronic processes, especially at high-energy proton–
antiproton colliders at CERN and Fermilab, probe the strong interactions at
extremely high Q2 (e. g., 104 GeV2). The observations are all consistent with

26
Fig. 8: Data on R = σ(e++e− → hadrons)/σ(e++e− → μ++μ−) as a function
of the center-of-mass energy W = √s, from S. Eidelman et al. [Particle Data
Group], Phys. Lett. B592, 1 (2004). The predictions of the quark parton model
and a fit to QCD are also shown.
the QCD predictions of underlying hard quark and gluon-scattering processes.
One test of QCD is to extract the coupling gs at various scales to see whether
it decreases at higher energy scales as predicted. The results, shown in Fig. 9,
are in spectacular agreement with the QCD expectations.
It is believed that at high temperatures and densities confinement would no
longer be relevant and that a plasma of quarks and gluons should be possible.
Presumably, quarks and gluons were unconfined in the very early Universe when
the temperature exceeded ∼ 1GeV. Experiments at the high energy heavy-ion
collider RHIC at the Brookhaven National Laboratory are attempting to
recreate such a state. They show possible indications, but the signatures are
not completely clear.
In view of these various successes, QCD is almost certainly the “correct”
theory of the strong interactions, even though there has been no single “gold-plated”
test. In fact, QCD is the only realistic candidate within the framework
of renormalizable field theories.
Hadronic Weak Interactions, Current Algebra, and Heavy
Quarks
The weak interactions, whether in the phenomenological local current–current
form or in the Weinberg–Salam [SU(2)×U(1)] form, need a specification of the
hadronic currents, both vector and axial. The experimental evidence showed
that the phenomenological form for the part of the current involving the proton

()αs

0.0100..23
Fig. 9: 1 Values of the strong 10GeV coupling αs = g2
102
s/4π from a variety of determina-tions
as a function of the scale μ at which they are measured, from S. Eidelman
et al. [Particle Data Group], Phys. Lett. B592, 1 (2004). The predictions of
QCD, with an overall scale determined from the data, are also shown.
and neutron was
¯n(x)[(0.9744 ± 0.0010)γ − (1.227 ± 0.004)γγ5] p(x) . (41)
This form bears a strong resemblance to the leptonic form in (16), and that,
paradoxically, is surprising, since, in general, form factors due to strong-interaction
corrections should appear. For the vector part, for example, the general expec-tation
would be that for low momentum transfers from proton to neutron (as
in beta decay)
hn|V|pi = GV (q2)¯nγp (42)
and the question is, Why should GV (0) be equal to unity? (The small deviation
of the vector coefficient in (41) from unity is due to the Cabibbo angle factor
discussed below.) R. P. Feynman and M. Gell-Mann, as well as S. Gershtein
and Y. B. Zel’dovich, pointed out that if the vector current were conserved,
and if R V0(x) d3x is normalized like the generator of a symmetry group, then
the observed result would follow. Feynman and Gell-Mann identified the vector
current with the current generating the isospin transformations. This satis-fied
the conditions that led to GV (0) = 1, and it led to a model-independent
characterization of the weak vector current. From this it was possible to predict

28
unambiguously the rate for the decay π+ → π0+e++νe, and certain “magnetic”
corrections to beta-decay spectra, all in excellent agreement with experiment.
In 1963, after the discovery of flavor SU(3), N. Cabibbo generalized this as-signment
and proposed that the weak current, now also describing the weak
interactions of the strange particles, has a form involving the SU(3)-generating
currents. The next important step was to give a general characterization of the
axial current. M. Gell-Mann proposed that there exists an additional global
symmetry of the strong interactions, also of the SU(3) type, but generated by
eight pseudoscalar charges whose associated currents are axial currents, in fact,
the weak axial currents. The difficulty that in the symmetric limit a pseu-doscalar
charge acting on a proton state, for example, yields another “odd”
proton, for which there is no experimental evidence, was resolved by Y. Nambu,
who made use of the Goldstone mechanism discussed earlier. The proposal
was that in the symmetry limit the axial current is conserved, but that the
solutions do not obey the symmetry. This leads to the prediction of massless
pseudoscalar particles, which would approximately describe the pions and their
SU(3) partners. The symmetry generated by the SU(3) generators Fi and the
pseudoscalar F5i is called a chiral [SU(3)× SU(3)] symmetry, and through the
study of weak interactions it has been established that it is a good symmetry,
leading to a number of experimentally verified relations involving matrix ele-ments
of the weak hadronic currents. In particular, in 1965 S. Adler and W.
Weisberger independently derived a relation between the coefficient GA(0) of
the axial current in (41) in terms of other observables, such as the πp cross
section. They obtained GA(0) ≃ 1.21, in excellent agreement with experiment.
The algebra SU(3) × SU(3) emerges quite naturally in a quark model. With
minimal couplings, the currents
¯q(x)γλiq(x) , ¯q(x)γγ5λiq(x) , (43)
where the λi are the SU(3) generalizations of the SU(2) Pauli matrices, are
conserved in the limit that the quark masses vanish. The current quark masses,
which are the masses appearing in the QCD Lagrangian (they may actually
be generated in the electroweak sector by the same Higgs mechanism which
yields the W and Z masses), break the symmetries associated with the axial
currents explicitly and generate small masses for the π, K, and η. Quark mass
differences break the vector symmetries and lead to multiplet mass differences.
From these effects one can estimate the current masses given in Table 2. The u
and d masses are extremely small. (The much larger constituent masses of order
300MeV/c2 in the naive quark model are dynamical masses associated with the
spontaneous breaking of chiral symmetry.) Experiments indicate mu6= md,
implying a breaking of isospin in the strong interactions, which is however no
larger than the (separate) electromagnetic breaking because of the small scale
of the masses. The much larger ms leads to a substantial breaking of SU(3).
The quark model gives a simple expression for the electromagnetic current of
the hadrons [Eq. (12)], with the coefficients determined by the quark charges.
Similarly, the weak neutral current coupling to the Z is given in (27). The weak

current that couples to the charged W is given (for three quarks) by
JW
= (¯u ¯ d¯s)QWγ(1 − γ5)
u
d
s

= (¯u ¯ d¯s)
0 0 0
cos θC 0 0
sin θC 0 0

γ(1 − γ5)
u
d
s

(44)
= ( ¯ d cos θC + ¯s sin θC)γ(1 − γ5)u .
This form leads to the Cabibbo theory, and the angle θC, the so-called Cabibbo
angle, is of magnitude 0.22 rad. Its origin lies in the difference in the ways
in which the strong and weak interactions break SU(3) symmetry. This is
associated with the quark masses, which are generated by the Higgs mechanism
in the standard model. Their values, as well as θC and the other mixing angles
introduced later, are free parameters that are not understood at present.
The part of the weak current involving s in (44) has the property that the
change in strangeness and the change in charge are equal (S = Q), in agree-ment
with experiment. The Cabibbo theory explains a large number of strange
particle decays with a universal choice of θC. It has one difficulty: If this
current is incorporated into a gauge theory of the weak interactions (e. g., the
SU(2) × U(1) model) in a manner analogous to the leptonic current, then the
neutral intermediate W0 vector meson couples naturally to a neutral current
obtained by commuting QW with its adjoint,
W] =
[QW,Q†
−1 0 0
0 cos2 θC sin θC cos θC
0 sin θC cos θC sin2 θC

. (45)
Among the neutral currents there will be strangeness-changing neutral currents
of the type ( ¯ ds + ¯sd) sin θC cos θC (we ignore the γ matrices for brevity), which
give rise to processes such as
K+ → π+ + ν + ¯ν ,
K0L
→ μ+ + μ−
(46)
at rates much larger than experimental limits or observations. Thus, a major
modification is needed. The solution had actually been proposed before the
Weinberg–Salam theory became popular. In 1970, S. Glashow, J. Iliopoulos,
and L. Maiani, building on some earlier work of Glashow and J. D. Bjorken,
proposed that the number of quark flavors be extended, with a fourth quark c
carrying a new conserved quantum number called charm. The c quark is taken
to have charge 2
3 , hypercharge 1
3 , and baryon number B = 1
3 . The weak current,

30
¯Fig. 10: K
Higher-order ¯K
weak diagram leading to the mass difference between
Ks ∼ K0+ 0 and KL ∼ K0− 0. Inclusion of the diagram with intermediate
t quarks can also account for the observed CP violation.
constructed to have the form
(¯u ¯ d¯s¯c)QWγ(1 − γ5)

u
d
s
c

(47)
with
QW =
0 0 0 0
cos θC 0 0 −sin θC
sin θC 0 0 cos θC
0 0 0 0

, (48)
treats the s quark symmetrically with the d. This implies that the neutral cur-rent
constructed as in (45) does not have any strangeness-changing (or charm-changing)
terms. The smallness of the Cabibbo angle implies that the dominant
charged-current transitions are u → d and c → s, which means that mesons in-volving
c quarks are predicted to usually decay into final states with strangeness
(e. g., a ¯K
) rather than into nonstrange (e. g., pions) final states.
Notice that if we set θC = 0 and replace (u, d, s, c) by (e−, νe, μ−, νμ), we get
the leptonic current. It was this analogy that originally led Bjorken and Glashow
to generalize the SU(3) flavor symmetry to the four-flavor SU(4) symmetry.
The analogy goes deep. It turns out that renormalizability demands that each
lepton pair [e. g., (e, νe)] must have a compensating quark pair [e. g., (u, d)], so
that the existence of the charmed quark is compelling in gauge theories. In
the early 1970s it was also realized that the observed mass difference between
the two neutral kaons could be accounted for by a calculable higher-order weak
effect (Fig. 10) if the c quark existed and had a mass around 1.5GeV/c2.
In 1974 in the experimental study of the ratio R in (40), B. Richter and
collaborators at the Stanford Linear Accelerator Center, simultaneously with S.
Ting and collaborators at Brookhaven National Laboratory, who were studying
the reaction
p + Be → e+ + e− + hadrons,

found an extremely sharp resonance at 3097MeV and, soon after that, another
one at 3685MeV (Fig. 8). It is interesting that these had been anticipated theo-retically
by T. Appelquist and D. Politzer, who advanced reasons why the J = 1
c¯c (charmed quark–antiquark) states should be quite long-lived. This interpre-tation
of the J/ψ(3097) and ψ(3685) as 1 3S1 and 2 3S1 charmonium states
was confirmed by the later discovery of 3P0, 3P1, 3P2, and 2S0 states as well
as additional 3S1 radial excitations in the vicinity of the ones already discov-ered.
There now exists a complete and well-studied charmonium spectroscopy.
The arrangement of levels and the spacing cannot be understood in terms of a
simple 1/r potential, as for positronium. Rather, a better fit is obtained with
a potential of the form V (r) = A/r + Br, where the first term represents the
short-range (asymptotically weakly coupled) contribution and the second term
represents the long-range confining potential. The extreme narrowness of the
3S1 resonances can be understood qualitatively by arguing that a c¯c → u¯u (say)
transition can only take place through the mediation of three color-carrying
gluons, and these couple weakly for large momentum transfers.
Numerous hadrons consisting of a single c quark bound to ordinary quarks
or antiquarks have also been identified and their decay modes studied. These
include the spin-0 (c¯u) D meson, with mass 1865MeV/c2, the corresponding
spin-1 D∗ meson of mass 2007MeV/c2, their charged (D+ = c ¯ d) partners, and
the D+
s (c¯s) at 1968MeV/c2, as well as charmed baryons with isospin 0, 1,
and 1
2 at 2285, 2455, and 2470MeV/c2, respectively. The spectroscopy of the
charmed hadrons, as well as additional evidence from deep inelastic neutrino
scattering, have thus clearly established the existence of the c quark as needed
for a consistent and realistic gauge theory of the weak interactions.
Similarly, a third (heavy) τ lepton was discovered at SLAC in 1976. The
observed weak interactions of the τ left little doubt of the existence of a third
(approximately) massless neutrino partner, but the interactions of the ν were
not observed directly until 2000 at Fermilab. Renormalizability then implied
that another quark pair, which we label (t, b) for top and bottom, respectively,
with charges 2
3 and −1
3 , should exist. This notion of lepton–quark symmetry
was confirmed in 1977 by the discovery by L. Lederman and collaborators at
the Fermi National Accelerator Laboratory of two new narrow resonances, the
(9 460MeV/c2) and ′(10 023MeV/c2), in the reaction
p + Be → + hadrons → μ+ + μ− + hadrons.
The interpretation of these states as 3S1 states of a new quarkonium composed
of b¯b
quarks with charge 1
3 was supported by their production in e+e− collisions
at DESY and at the Cornell Electron Storage Ring (CESR), where additional
3S1 as well as P wave states were also discovered. The bottom mesons B+ = u¯b
and B0 = d¯b
, with masses ∼5279MeV/c2, as well as their antiparticles and
many other bottom mesons and baryons, have also been identified and studied
in detail at a number of laboratories.
The observed weak interactions of the b quark as well as renormalizability
and lepton–quark symmetry indicated that a top quark (t) should exist. Fur-thermore,
by the early 1990’s indirect arguments based on the consistency of

32
weak neutral current and Z-pole data, which are sensitive to a heavy t quark
mass through higher-order corrections, predicted a mass mt = 150±30GeV/c2.
The t was finally discovered at the Tevatron in 1994 in the expected range. It
is too short-lived to form observable bound states. Rather, pairs of t and ¯t
are produced through the strong interactions, and the direct decays into b (¯b
)
and additional quark jets or leptons are observed. The current indirect predic-tion
mt = 172+10
−7 GeV/c2 is in excellent agreement with the observed value of
178.0 ± 4.3GeV/c2, dramatically confirming the standard electroweak model.
Why the t is so much heavier than the other fermions is not understood.
In the four-quark model, QW in (47) involves a single mixing angle θC ≃ 0.22,
and the higher-order diagram in Fig. 10 can account for the KL − KS mass
difference. The generalization of QW to six quarks involves three mixing angles
and one complex phase. In addition to θC, the dominant b quark transition
b → c is described by a mixing angle θbc ≃ 0.04 (determined from the relatively
long b lifetime). The much weaker b → u transition is characterized by the angle
θbu ≃ 0.004. These angles suffice to account for the large mixing observed at
DESY between the B0 = d¯b
and ¯B
0 = d¯b
states, by a second-order diagram
analogous to Fig. 10 (with the external s replaced by b). Large mixing between
the B0
s = s¯b
and ¯B
0
s = ¯sb is predicted.
Even after the discovery that the weak interactions violated parity (P) and
¯K
charge conjugation (C) invariance, it was generally believed that CP invariance
was maintained. In particular, the CP = +1 state KS ∼ K0 + 0 should
¯decay rapidly K
(τ ∼ 0.89 × 10−10 s) into π+π− or π0π0, while the CP = −1
state KL ∼ K0 − 0 should decay into 3π in ∼ 5.2 × 10−8 s. However, in
1964 J. Cronin, V. Fitch, and collaborators working at Brookhaven observed
the CP violating decays KL → 2π with branching ratios of ∼ 10−3. The results
could be accounted for by a small CP violating mixing between the KL and
KS states. One possibility was that this mixing is generated by an entirely
new superweak S = 2 interaction. In 1973, M. Kobayashi and M. Maskawa
suggested that CP breaking could be generated by the higher-order diagram
in Fig. 10 if there were three (or more) fermion families, implying observable
phases in QW. The observed KL − KS mixing could thus be generated either
by the three-family standard model or by the superweak model. However, in
the 1990’s experiments at CERN and Fermilab observed small differences in the
CP violating KL → π0π0 and π+π− rates (relative to KS,) that required direct
CP breaking in the decay amplitude, not just in the state mixing, strongly
confirming the standard SU(2) × U(1) model interpretation.
Subsequently, there have been very detailed studies of CP-violating asym-metries,
rare decays, and other aspects of B meson decays at a number of
laboratories, especially the “B factories” at CESR, SLAC and KEK (Japan).
All data is consistent with the standard model, in particular with the interpre-tation
that the patterns of decays and CP-violation can be accounted for by the
unitary Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix QW. This is
illustrated by the unitarity triangle shown in Fig. 11. The charged current sec-tor
will be tested even more stringently in the future following the (probable)
measurement of B0
s
¯B
0
s mixing and more precise measurements of rare decays

and CP-violating asymmetries in B meson decays.
1.5
1
0.5
0
-0.5
-1
-1.5
sin 2b
a
|Vub/Vcb|
a
Dmd
DmsDmd
eK
eK
a
g b
-1 -0.5 0 0.5 1 1.5 2
r
h
excluded area has CL 0.05
CKM
f i t t e r
ICHEP 2004
Fig. 11: The current status of the CKM matrix, from J. Charles et al. [CKM-fitter
Group Collaboration], hep-ph/0406184 and ckmfitter.in2p3.fr. ¯ρ and ¯η
parametrize elements of the matrix. Various observations from K and B decays
determine the shaded bands. They are all consistent with each other within the
region surrounding the vertex (labeled by angle α) of the unitarity triangle.
Neutrino Mass and Oscillations
For many years all experimental searches for nonzero neutrino masses yielded
negative results, implying that they are either massless or much lighter than
the charged leptons and quarks. The original SU(2) × U(1) standard model
was constructed to yield massless ν’s, but most theoretical ideas for more com-plete
theories predicted that m would be nonzero at some level. The first
experimental hints of nonzero mass were from the Homestake Solar Neutrino
Experiment in the late 1960’s. R. Davis and collaborators observed events in
a large underground chlorine experiment that were later confirmed to be in-duced
by electron neutrinos (νe) produced by nuclear reactions in the core of
the Sun. However, they only observed about one third of the rate predicted by
J. Bahcall and collaborators based on the observed Solar luminosity and theo-

34
retical models of the Sun. This Solar neutrino problem was later confirmed by
other experiments using gallium and water-based detectors. One explanation
was that the νe were oscillating (converting) into other types of neutrino (νμ or
ν ), to which the experiments were insensitive, through effects associated with
(tiny) neutrino masses and mixings (analogous to the mixings observed for the
quarks). However, there was the alternative possibility that the astrophysical
uncertainties in the theoretical calculation had been underestimated. The situ-ation
was resolved in 2002 when the Sudbury Neutrino Observatory (Canada)
was able to measure the νe and total fluxes independently using a heavy water
detector, with the result that the νe really were converting to νμ or ν . This was
later confirmed by the observation by the KamLAND experiment (Japan) of the
disappearance of ¯νe’s produced in power reactors. Prior to the Sudbury results,
a different type of atmospheric neutrino oscillation, of νμ’s produced by cosmic
ray interactions in the atmosphere into (presumably) ν ’s, was established by
the Super-Kamiokande experiment (Japan).
There are three types of neutrinos, and neutrino oscillations presumably in-volve
three mixing angles and one or more leptonic CP-violating phases, similar
to the CKM quark mixing. However, a simplified description assumes that only
two states are relevant for a given type of experiment. For example, suppose
that the νe and νμ, the states which are associated with the e− and μ−, are
superpositions of states ν1,2 of definite masses m1,2,
νe = ν1 cos θ + ν2 sin θ ,
νμ = −ν1 sin θ + ν2 cos θ . (49)
Then if an initially produced νe has energy E large compared to m1,2 the mixing
effect may cause it to transform into νμ after travelling a distance L, with
probability
P(νe → νμ;L) = sin2 2θ sin2 1.27m2L
E
, (50)
where m2 = m22
− m21
is expressed in eV/c2, L in m, and E in MeV. The
observed Solar and atmospheric oscillation parameters (which correspond to
different m2 and θ) are shown in Fig. 12. Note that the mass scales are tiny
compared to the other fermions, but that the mixing angles are large, unlike the
small quark mixings (there is a third neutrino mixing angle that is consistent
with zero).
The neutrino masses and mixings are interesting because they are so dif-ferent,
and they may be an indication of new physics underlying the standard
model at much shorter distance scales (i. e., much larger mass scales). For exam-ple,
there are various seesaw models which predict very small neutrino masses
m ∼ m2
D/M ≪ mD, where mD is comparable to the quark and charged lepton
masses and M ≫ mD is the new physics scale (e. g., 1014 GeV/c2). Another
interesting possibility is that the neutrinos are Majorana, which means that the
mass effects can convert neutrinos into antineutrinos so that the total lepton
number (number of leptons minus the number of antileptons) is not conserved.

Fig. 12: Current status of neutrino oscillations, from H. Murayama,
http://hitoshi.berkeley.edu/neutrino. The regions for m2 ∼ 2 × 10−3 eV2
and tan2 θ ∼ 1, and for m2 ∼ 8 × 10−5 eV2 and tan2 θ ∼ 0.4 are indicated by
the atmospheric and Solar neutrino oscillations, respectively. (The two regions
represent different combinations of neutrino states.)
This will be probed in neutrinoless double beta decay (ββ0 ) experiments search-ing
for rare nuclear decays (Z,N) → (Z + 2,N − 2) + e− + e−, (i. e., without
the two antineutrinos expected from two successive β decays). Another out-standing
issue is the absolute neutrino mass scale, since oscillations only probe
mass-square differences. There is an upper limit of ∼ 0.2 eV/c2 from cosmology
(otherwise, relic neutrinos left over from the big bang would modify the forma-tion
of structure in the Universe). This will be refined in the future, and ββ0
and kinematic effects in ordinary β decay may also be important. More detailed
information on the spectrum, mixings, and possible leptonic CP violation are
also anticipated, as is clarification of possible indications of additional types of
oscillations that cannot be accommodated in the three-neutrino framework.

36
Problems with the Standard Model
The standard model (QCD plus the Weinberg–Salam electroweak model and
general relativity) has been spectacularly successful. Although the elementary
Higgs mechanism for symmetry breaking has not yet been tested and may possi-bly
be replaced by a dynamical mechanism, the basic structure of the standard
model is almost certainly correct at some level. However, it contains far too
much arbitrariness to be the final story of Nature. One way of seeing this is
that the minimal version has 27 free parameters (29 for Majorana neutrinos),
not including electric charges. In addition, the standard model suffers from:
(a) The Gauge Problem: The standard model gauge group is a complicated
direct product of three factors with three independent coupling constants.
Charge quantization, which refers to the fact that the magnitudes of the
proton and of the electron electric charges are the same, is not explained.
(b) Fermion Problem: The standard model involves a very complicated re-ducible
representation for the fermions. Ordinary matter can be con-structed
out of the fermions of the first family (νe, e−, u, d). We have no
fundamental understanding of why the additional families (νμ, μ−, c, s),
(ν , τ, t, b), which appear to be identical with the first except that they
are heavier, exist. In addition, the standard model does not explain or
predict the pattern of fermion masses, which are observed to vary over
many orders of magnitude, or mixings. Also, the CP violation associated
with quark mixing is not sufficient to explain the generation of the excess
of matter compared to antimatter in the universe (the baryon asymmetry),
therefore requiring the existence of some additional form of CP violation.
(c) Higgs/Hierarchy Problem: The spontaneous breakdown of the SU(2) × U(1) symmetry in the standard model is accomplished by the introduction
of a Higgs field. Consistency requires that the mass of the Higgs boson be
not too much different from the weak interaction scale; that is, it should
be equal to the W mass within one or two orders of magnitude. However,
there are higher-order corrections which change (renormalize) the value of
the square of the Higgs mass by δm2
m2
H ∼ p, where mp = (GN/~c)−1/2 ≃ 1019 GeV/c2 is the Planck (gravity) scale. Therefore, δm2
H/M2W
≥ 1034,
and the bare value of m2
H in the original Lagrangian must be adjusted
or fine-tuned to 34 decimal places. Such a fine-tuning is possible, but
extremely unattractive.
(d) Strong CP Problem: It is possible to add an additional term, characterized
by a dimensionless parameter θ, to the QCD Lagrangian which breaks P,
T, and CP invariance. Limits on the electric dipole moment of the neutron
require θ to be less than 10−9. However, weak interaction corrections
change or renormalize the lowest-order value of θ by about 10−3 – that
is, 106 times more than the total value is allowed to be. Again, one must
fine-tune the bare value against the correction to a high degree of precision.

(e) Graviton Problem: The graviton problem has several aspects. First, grav-ity
is not unified with the other interactions in a fundamental way. Second,
even though general relativity can be incorporated into the model by hand,
we have no idea how to achieve a mathematically consistent theory of
quantum gravity: attempts to quantize gravity within the standard model
framework lead to horrible divergences and a nonrenormalizable theory.
Finally, there is yet another fine-tuning problem associated with the cos-mological
constant. The vacuum energy density hV i associated with the
spontaneous symmetry breaking of the SU(2) × U(1) model generates
an effective renormalization of the cosmological constant δ = 8πGnhV i,
which is about 50 orders of magnitude larger than the observed limit (or
value, if the observed dark energy or acceleration of the universe is due
to a cosmological constant). One must fine-tune the bare cosmological
constant against the correction to this incredible degree of precision.
Extensions of the Standard Model
There must almost certainly be new physics beyond the standard model. Some
of the possible types of new physics that have been discussed extensively in
recent years are shown in Table 6. Additional gauge bosons, fermions, or Higgs
bosons do not by themselves solve any problems, but may exist at accessible
energies as remnants of underlying physics such as unified gauge groups or super-strings.
The Z-pole data from CERN and SLAC, as well as the experimental
evidence from the cosmological abundance of helium (interpreted within the
framework of the standard “big-bang” cosmological model), implies that there
are no additional fermion families beyond the three already known, unless the
associated neutrinos are very heavy (greater than ≃ 45GeV/c2). More likely
are exotic heavy fermions, predicted by many theories, which are not simply
repetitions of the known families. New gauge interactions, especially heavy
electrically neutral Z′ bosons analogous to the Z in (26) are predicted by many
extensions, as are extended Higgs sectors.
Family symmetries are new global or gauge symmetries which relate the
fermion families. Another approach to understanding the fermion spectrum is to
assume that the quarks and leptons are bound states (composites) of still smaller
constituents. However, experimental searches for substructure suggest that the
underlying particles must be extremely massive ( 1000GeV/c2). Hence, un-like
all previous layers of matter, extremely strong binding would be required.
Neither of these ideas has led to a particularly attractive model.
The fine-tuning problem associated with the Higgs mechanism could be solved
if the elementary Higgs fields were replaced by some sort of bound-state mech-anism.
However, models which generate fermion as well as W,Z masses are
complicated and tend to predict certain unobserved decays at too rapid a rate
and lead to unacceptably large corrections to precision electroweak data. Lit-tle
Higgs models are a variant in which the elementary Higgs is replaced by
the approximately massless Goldstone boson of a new global symmetry, which
ensures cancellation between different higher-order corrections to the square of

38
Table 6: Some possible extensions of the standard model.
Typical
Model scale (GeV) Motivation
New Ws, Zs, fermions, Higgs 102–1019 Remnant of something else
Family symmetry 102–1019 Fermion (No compelling models)
Composite fermions 102–1019 Fermion (No compelling models)
Composite Higgs 103–104 Higgs (No compelling models)
Composite W, Z (G, γ?) 103–104 Higgs (No compelling models)
Little Higgs 103–104 Higgs
Large extra dimensions (d 4) 103–106 Higgs, graviton
New global symmetry 108–1012 Strong CP
Kaluza–Klein 1019 Graviton
Higgs (0) ⇔ gauge (1) ⇔ Graviton (2) (d 4)
Grand unification 1014–1019 Gauge
Strong ⇔ electroweak
Supersymmetry/supergravity 102–104 Higgs, graviton
Fermion ⇔ boson
Superstrings 1019 All problems!?
Strong ⇔ electroweak ⇔ gravity
Fermion ⇔ boson (d 4)
the Higgs mass. Composite W and Z bosons could be an alternative to spon-taneous
symmetry breaking, but such theories abandon most of the advantages
of gauge theories. The strong CP problem could be resolved by the addition
of a new global symmetry which ensures that θ is a dynamical variable which
is zero in the lowest energy state. Such models imply a weakly coupled Gold-stone
boson (the axion) associated with the symmetry breaking. Constraints
from astrophysics and cosmology limit the symmetry breaking scale to the range
108–1012 GeV.
Kaluza–Klein theories are gravity theories in d 4 dimensions of space and
time. It is assumed that four dimensions remain flat while the other d − 4
are compactified or curled up into a compact manifold with a typical radius of
~/mpc ∼ 10−33 cm. Gravitational interactions associated with these unobserved
compact dimensions would appear as effective gauge interactions and Higgs

Fig. 13: The momentum dependent normalized coupling constants g3 ≡ gs,
g2 ≡ g,and g1 ≡ p5/3g′.
particles in our four-dimensional world. Although such ideas are extremely
attractive for unifying the interactions they have great difficulty in incorporating
parity violation and also in achieving a stable configuration for the compact
manifold. Many of the aspects have reemerged more successfully within the
framework of superstrings.
Similarly, theories of large extra dimensions postulate the existence of one
or more extra dimensions of space that could be as large as a fraction of a
mm! There are many variants on these ideas, but one promising version as-sumes
that gravitons (the spin-2 quanta of the gravitational field) can prop-agate
freely in the bulk of the extra dimensions, while the ordinary particles
are somehow confined to the ordinary 3-dimensional space (the brane). This
eliminates the Higgs/hierarchy problem because the fundamental (largest) mass
scale of nature mf can be far smaller (e. g., 105 GeV/c2) than the Planck scale
mp ∼ 1019 GeV/c2, with the apparent weakness of gravity due to the fact that
the 1/r2 force law is modified for distances smaller than the size of the extra
dimension. Such ideas may emerge as a limiting case of superstring theories
(which involve extra dimensions), but introduce a new hierarchy problem, viz.,
why is the large dimension much larger than the expected ~/mf c?
Much work has been devoted to grand unified gauge theories (GUTs), in
which it is proposed that at very high energies (or very short distances) the un-derlying
symmetry is a single gauge group that contains as its subgroups both
SU(3) for color and SU(2) × U(1) for the weak and electromagnetic interac-tions.
The larger symmetry is manifest above a unification scale MX at which
it is spontaneously broken. At lower energies, the symmetry is hidden and the
strong and electroweak interactions appear different. The unification scale can
be estimated from the observed low-energy coupling constants. The energy-dependent
couplings are expected to meet at MX, as in Fig. 13. Since they vary
only logarithmically, MX is predicted to be extremely large. Two versions of

40
GUTs are typically considered, one in which only the standard model particles
exist below MX and contribute to the variation of the couplings, and one in
which a new class of superpartners and an extra Higgs multiplet expected in
supersymmetry (discussed below) are included. The predicted values of MX
are typically around 1014 GeV/c2 and 3 × 1016 GeV/c2 for these two cases,
respectively. (These heady speculations rest on the assumption that nothing
fundamentally different occurs in a leap in energy scale from 1 to MX. This
is extremely unlikely, but then who would have thought that Maxwell’s equa-tions
are good over distances ranging from astronomical down to 10−15 cm?)
From the strong and electromagnetic couplings one can predict the weak angle
sin2 θW. The simplest non-supersymmetric GUTs predict a value ∼ 0.20, well
below the present precisely known value, while the supersymmetric ones are in
reasonable agreement with experiment, especially when uncertainties from mass
differences at low energy and MX are taken into account.
In addition to unifying the interactions, the additional symmetries within
a grand unified theory relate quarks, antiquarks, leptons, and antileptons. For
example, the simplest grand unified theory – the 1974 SU(5) model of H. Georgi
and S. Glashow – assigns the left-helicity (i. e., spin oriented opposite to mo-mentum)
fermions of the first family to a reducible 5*+ 10-dimensional repre-sentation:
W± l
5∗ 10

νe
¯ d
e−


u
e+ ¯u
d

← X, Y → ↔ X, Y ↔
(51)
The neutrino and electron can be rotated into the anti-down quark by the new
symmetry generators, as can the positron and the up, down, and anti-up quarks.
Because of this relation between the different types of fermions, the grand unified
theories naturally explain charge quantization. In addition, there are new gauge
bosons associated with these extra symmetries. The SU(5) model contains new
bosons which carry both color and electric charge, known as the X and Y
bosons, with masses MX ∼ MY ∼ 1014 GeV/c2 (3 × 1016 GeV/c2) for the non-supersymmetric
(supersymmetric) case. These can mediate proton decay (and
also the decay of otherwise stable bound neutrons). A diagram for p → e+π0
is shown in Fig. 14. Motivated by such predictions, a number of experiments
have searched for proton decay. No events have been observed, and one finds
τ(p → e+π0) 5 × 1033 yr, in conflict with the prediction 4 × 1029±2 yr of the
simplest non-supersymmetric GUTs. The predicted lifetime is much longer in
the supersymmetric case because of the larger MX. However, there are other
diagrams involving the superpartners in those models that lead to such modes
as p → K+¯ν, with typical lifetimes around 1033 yr. The simplest such models
are excluded by the corresponding experimental bounds, but some extended or
nonminimal models are still viable.
One of the mysteries of nature is the cosmological baryon asymmetry. It is an
observational fact that our part of the Universe consists of matter and not an-timatter:
there is approximately one baryon for every 1010 microwave photons,

Fig. 14: A typical proton decay diagram.
but essentially no antibaryons. This could not be due to initial conditions if the
Universe underwent a period of extremely rapid expansion (inflation), which
many cosmologists believe is the most likely explanation of the almost exact
flatness and homogeneity of the Universe. At one time it was believed that
in GUTs this asymmetry could be generated dynamically in the first instant
(the first 10−35 s or so) after the big bang, by baryon number violating inter-actions
related to those which lead to proton decay. This now appears unlikely
because the simplest GUTs predict that equal baryon and lepton asymmetries
would have been generated, and these would later have been erased due to
non-perturbative baryon and lepton number violating processes associated with
the electroweak theory prior to the electroweak phase transition, i. e., when the
temperature was higher than the electroweak scale. (A subsequent period of
inflation would also have wiped out the asymmetry.) However, alternative ideas
have emerged for dynamically generating the asymmetry. These include the
possibility of first generating the lepton asymmetry by heavy particle decays in
seesaw models of Majorana neutrino mass (leptogenesis), with some of the lep-ton
asymmetry converted to a baryon asymmetry by the nonperturbative effects;
or the possibility of actually generating the asymmetry during the electroweak
transition, especially in extensions of the supersymmetric standard model.
Grand unified theories predict the existence of superheavy magnetic monopole
states with masses MM ∼ MX/αG ∼ 1016−1018 GeV/c2. These may have been
produced during phase transitions during the early Universe. If they were still
left over as relics today they would contribute far more mass to the present Uni-verse
than is allowed by observations. However, they would have been diluted
to a negligible density if the Universe underwent a period of inflation.
Grand unified theories offer little or no help with the fermion, Higgs, strong
CP, or graviton problems, and the simplest non-supersymmetric versions are
ruled out by sin2 θW and proton decay. GUTs are probably not ambitious
enough, but it is likely that some ingredients (e. g., related to charge quantiza-tion)
will survive. It is also possible that there is an underlying grand unification
in a theory with extra space dimensions, such as a superstring theory, but the
full GUT symmetry is not manifest in the three large dimensions of space.
Supersymmetry (see Supersymmetry and Supergravity) is a new kind of sym-

42
metry in which fermions are related to bosons. Realistic models are complicated
in that they require more than a doubling of the number of fundamental par-ticles.
For example, one must introduce a spin-0 superpartner of each known
fermion, a spin- 1
2 partner of each gauge boson or Higgs field, etc. The primary
motivation is the Higgs problem: higher-order corrections associated with the
new particles cancel the unacceptable renormalization of the Higgs mass that
occurs in the standard model. If supersymmetry were exact, the new particles
would be degenerate with the ordinary particles. However, there exist experi-mental
lower limits of order 100−200GeV/c2 on the masses of most superpart-ners,
so that supersymmetry (if it exists) must be broken. Some models assume
that supersymmetry is broken at some very large scale (e. g., 1011 GeV) in a sec-tor
which is only coupled very weakly (by gravitational-strength interactions) to
the ordinary particles and their partners, implying superpartner masses in the
range 102–104 GeV/c2. (Larger masses would fail to solve the Higgs problem.)
Variant versions break the supersymmetry at a lower scale (e. g., 105 GeV) in a
sector that is somewhat more strongly coupled (by gauge interactions) to the
ordinary sector, again leading to 102–104 GeV/c2 superpartner masses. In both
cases, the superpartner mass scale actually determines the scale of electroweak
symmetry breaking.
Many supersymmetric theories involve a conserved R-parity, which distin-guishes
the ordinary particles from their superpartners. This implies that the
lightest superpartner (the LSP) is stable. If it is electrically neutral, such as
the spin- 1
2 partner of a neutral gauge or Higgs field, it would be an excellent
candidate for the cold dark matter of the Universe. (An alternate possibility
if R-parity is not conserved would be the axion described above. Massive neu-trinos
in the eV/c2 range are not good candidates because they are too light,
and would prevent mass from clustering on small scales.) As described in the
GUT section, supersymmetric grand unified theories lead to larger values for
sin2 θW (in better agreement with experiment), and a larger unification scale,
reducing but not necessarily solving the problem that proton decay has not been
observed.
Just as the promotion of an ordinary internal symmetry to a gauge symmetry
implies the existence of spin-1 gauge bosons, the requirement that supersym-metry
transformations can be performed independently at different space-time
points implies the existence of spin-2 gravitons; i. e., gauged supersymmetry
automatically unifies gravity. However, supergravity theories do not solve the
problems of quantum gravity. Higher-order corrections are still divergent and
nonrenormalizable. There is no experimental evidence for supersymmetry, but
if it exists at the scales suggested by the Higgs problem it should be observed
in the near future at the LHC pp collider being constructed at CERN. Whether
Nature chooses to make use of supersymmetry remains to be seen.
Superstrings are a very exciting development which may yield finite theories
of all interactions with no free parameters. Superstring theories introduce new
structure around the Planck scale. The basic idea is that instead of working
with (zero-dimensional) point particles as the basic quantities, one considers
one-dimensional objects known as strings, which may be open (e. g., ending on

membrane-like objects known as branes) or closed. The quantized vibrations of
these strings lead to an infinite set of states, the spectrum of which is controlled
by the string tension, given by the square of the Planck scale τ ∼ m2
p. When
one probes or observes a string at energies much less than the Planck scale, one
sees only the “massless” modes, and these represent ordinary particles. The
physical size of the string is given by the inverse of the Planck scale ≃ 10−33 cm,
and at larger scales a string looks like a point particle. There are actually 5
types of consistent string theories, but it is now believed that these (and one 11-
dimensional supergravity theory) are limiting cases of an even more fundamental
(and mysterious) M theory.
The mathematical consistency of superstring theories requires (in most ver-sions)
that there are nine dimensions of space and one of time. Presumably,
the extra six space dimensions are curled into a compact manifold of radius
∼ 10−33 cm, reminiscent of Kaluza–Klein theories. (It is possible that one or
more or the dimensions have much larger sizes, leading to a realization of the
large extra dimension theories described earlier.) In the most realistic closed
string case the interactions are due to gauge interactions in the ten-dimensional
space. The absence of mathematical pathologies requires an essentially unique
gauge group called E8 × E8. At energy scales less than the Planck scale an
effective particle field theory in four dimensions emerges, with a supersymmet-ric
gauge symmetry based on a subgroup of E8 × E8. The effective group, the
number of fermions, and their masses, mixings, etc., are all determined by the
way in which the extra dimensions are compactified. In the open string theories
the gauge and other interactions are determined by the configurations of the
branes in the nine space dimensions.
In principle, superstring theories have no arbitrary parameters or other fea-tures,
and most likely they yield completely finite (not just renormalizable)
quantum theories of gravity and all the other interactions. That is, they are
candidates for the ultimate “theory of everything.” However, there are enor-mous
numbers of possible ways in which the extra dimensions can compactify,
and at present we do not possess the principles or mathematical tools to deter-mine
which is chosen. It is therefore not clear what the predictions of superstring
theories really are (e. g., whether they lead to an effective supergravity GUT)
or whether they correspond to the real world. In fact, there have been recent
speculations that there is no selection principle, and that the choice of possible
compactifications is essentially random out of an enormously large landscape of
possible vacua. There may be different physics occurring in different regions of
a very large chaotic Universe. In that case it is possible that supersymmetry
is broken at a high scale, and the Higgs hierarchy problem is apparently solved
by a fine-tuning, perhaps related to anthropic selection principles, i. e., that life
could only develop in the subset of worlds for which the“fine-tuning” occurred.
The Future
The standard model is extremely successful, but there is almost certainly new
physics underlying it. There are many theoretical ideas concerning this new

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